# How do you simplify \frac{(3ab)^2(4a^3b^4)^3}{(6a^2b)^4}?

Nov 2, 2014

$\frac{{\left(3 a b\right)}^{2} {\left(4 {a}^{3} {b}^{4}\right)}^{3}}{{\left(6 {a}^{2} b\right)}^{4}}$

since $4 = {2}^{2}$ and $6 = 3 \cdot 2$,

$= \frac{{\left(3 a b\right)}^{2} {\left({2}^{2} {a}^{3} {b}^{4}\right)}^{3}}{{\left(3 \cdot 2 {a}^{2} b\right)}^{4}}$

by distributing the powers,

$= \frac{{3}^{2} {a}^{2} {b}^{2} \cdot {2}^{6} {a}^{9} {b}^{12}}{{3}^{4} {2}^{4} {a}^{8} {b}^{4}}$

by collecting like factors,

$= \frac{{3}^{2} {2}^{6} {a}^{11} {b}^{14}}{{3}^{4} {2}^{4} {a}^{8} {b}^{4}}$

By cancelling out ${3}^{2}$, ${2}^{4}$, ${a}^{8}$, and ${b}^{4}$,

$= \frac{{2}^{2} {a}^{3} {b}^{10}}{3} = \frac{4}{3} {a}^{3} {b}^{10}$

I hope that this was helpful.